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DC (LK/SW) 101619/2© UCLES 2015 [Turn over

Cambridge International ExaminationsCambridge International General Certificate of Secondary Education

*9205519979*

maThEmaTICs 0580/43

Paper 4 (Extended) may/June 2015

2 hours 30 minutes

Candidates answer on the Question Paper.

Additional Materials: Electronic calculator Geometrical instruments Tracing paper (optional).

REaD ThEsE INsTRUCTIONs FIRsT

Write your Centre number, candidate number and name on all the work you hand in.Write in dark blue or black pen.You may use an HB pencil for any diagrams or graphs.Do not use staples, paper clips, glue or correction fluid.DO NOT WRITE IN ANY BARCODES.

Answer all questions.If working is needed for any question it must be shown below that question.Electronic calculators should be used.If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place.For π, use either your calculator value or 3.142.

At the end of the examination, fasten all your work securely together.The number of marks is given in brackets [ ] at the end of each question or part question.The total of the marks for this paper is 130.

The syllabus is approved for use in England, Wales and Northern Ireland as a Cambridge International Level 1/Level 2 Certificate.

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1y

x

6

5

4

3

2

1

–1

–2

–3

–4

–5

–6

0–1 1 2 3 4 5 6 7–2–3–4–5

T

U

V

(a) On the grid, draw the image of

(i) triangle T after a reflection in the line x = –1, [2]

(ii) triangle T after a rotation through 180° about (0, 0). [2]

(b) Describe fully the single transformation that maps

(i) triangle T onto triangle U,

Answer(b)(i) ......................................................................................................................................

...................................................................................................................................................... [2]

(ii) triangle T onto triangle V.

Answer(b)(ii) .....................................................................................................................................

...................................................................................................................................................... [3]

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2 (a) (i) Eduardo invests $640 at a rate of 2% per year compound interest.

Show that, at the end of 6 years, Eduardo has $721, correct to the nearest dollar.

Answer(a)(i)

[2]

(ii) Manuela also invests $640. At the end of 4 years, Manuela has $721.

Find the yearly compound interest rate.

Answer(a)(ii) ............................................. % [4]

(b) Carlos buys a motor scooter for $1200. Each year the value of the scooter decreases by 10% of its value at the beginning of that year.

Find the value of the scooter after 3 years.

Answer(b) $ ................................................. [2]

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3 ( )f x x82

= +2x

, x 0! .

(a) Complete the table of values for f(x).

x –5 –4 –3 –2 –1.5 1.5 2 2.5 3 3.5 4 5

f(x) –2.2 –1.5 –0.6 2.8 4.3 2.5 2.4 2.4 2.8

[3]

(b) On the grid, draw the graph of y = f(x) for –5 x –1.5 and 1.5 x 5.

y

x

5

4

3

2

1

–1

–2

–3

0 1 2 3 4 5–1–2–3–5 –4

[5]

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(c) Solve f(x) = 0.

Answer(c) x = ................................................. [1]

(d) By drawing a suitable line on the grid, solve the equation f(x) = 1 – x.

Answer(d) x = ................................................. [3]

(e) By drawing a tangent at the point (–3, –0.6), estimate the gradient of the graph of y = f(x) when x = –3.

Answer(e) ................................................. [3]

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4 The table shows the times, t minutes, taken by 200 students to complete an IGCSE paper.

Time (t minutes) 40 t 60 60 t 70 70 t 75 75 t 90

Frequency 10 50 80 60

(a) By using mid-interval values, calculate an estimate of the mean time.

Answer(a) .......................................... min [3]

(b) On the grid, draw a histogram to show the information in the table.

t40 50 60

Time (minutes)70 80 900

20

18

16

14

12

10

8

6

4

2

Frequencydensity

[4]

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5

A A A A B B C

(a) One of these 7 cards is chosen at random.

Write down the probability that the card

(i) shows the letter A, Answer(a)(i) ................................................. [1]

(ii) shows the letter A or B, Answer(a)(ii) ................................................. [1]

(iii) does not show the letter B. Answer(a)(iii) ................................................. [1]

(b) Two of the cards are chosen at random, without replacement.

Find the probability that

(i) both show the letter A,

Answer(b)(i) ................................................. [2]

(ii) the two letters are different.

Answer(b)(ii) ................................................. [3]

(c) Three of the cards are chosen at random, without replacement.

Find the probability that the cards do not show the letter C.

Answer(c) ................................................. [2]__________________________________________________________________________________________

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6 (a)

E D

F

A B

C

140°120°NOT TOSCALE

In the hexagon ABCDEF, AB is parallel to ED and AF is parallel to CD. Angle ABC = 90°, angle CDE = 140° and angle DEF = 120°.

Calculate angle EFA.

Answer(a) Angle EFA = ................................................. [4]

(b)

D

A B

C

X

30°

100°

NOT TOSCALE

In the cyclic quadrilateral ABCD, angle ABC = 100° and angle BDC = 30°. The diagonals intersect at X.

(i) Calculate angle ACB.

Answer(b)(i) Angle ACB = ................................................. [2]

(ii) Angle BXC = 89°.

Calculate angle CAD.

Answer(b)(ii) Angle CAD = ................................................. [2]

(iii) Complete the statement.

Triangles AXD and BXC are ............................................................. . [1]

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(c)

S

P

R

Q

Y

NOT TOSCALE

P, Q, R and S lie on a circle. PR and QS intersect at Y. PS = 11 cm, QR = 10 cm and the area of triangle QRY = 23 cm2.

Calculate the area of triangle PYS.

Answer(c) .......................................... cm2 [2]

(d) A regular polygon has n sides. Each exterior angle is equal to n

10 degrees.

(i) Find the value of n.

Answer(d)(i) n = ................................................. [3]

(ii) Find the size of an interior angle of this polygon.

Answer(d)(ii) ................................................. [2]

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7 (a) The total surface area of a cone is given by the formula A = πrl + πr2.

(i) Find A when r = 6.2 cm and l = 10.8 cm.

Answer(a)(i) .......................................... cm2 [2]

(ii) Rearrange the formula to make l the subject.

Answer(a)(ii) l = ................................................. [2]

(b) (i) Irina walks 10 km at 4 km/h and then a further 8 km at 5 km/h.

Calculate Irina’s average speed for the whole journey.

Answer(b)(i) ........................................ km/h [3]

(ii) Dariella walks x km at 5 km/h and then runs (x + 4) km at 10 km/h. The average speed of this journey is 7 km/h.

Find the value of x. Show all your working.

Answer(b)(ii) x = ................................................. [5]

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(c) (i) Priyantha sells her model car for $19.80 at a profit of 20%.

Calculate the original price of the model car.

Answer(c)(i) $ .................................................. [3]

(ii) Dev sells his model car for $x at a profit of y %.

Find an expression, in terms of x and y, for the original price of this model car. Write your answer as a single fraction.

Answer(c)(ii) $ .................................................. [3]

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8 (a) A cylindrical tank contains 180 000 cm3 of water. 45 cm

NOT TOSCALE

The radius of the tank is 45 cm.

Calculate the height of water in the tank.

Answer(a) ........................................... cm [2]

(b)

A B

CD

50 cm

70 cm

150 cm

NOT TOSCALE

40 cm

The diagram shows an empty tank in the shape of a horizontal prism of length 150 cm. The cross section of the prism is an isosceles trapezium ABCD. AB = 50 cm, CD = 70 cm and the vertical height of the trapezium is 40 cm.

(i) Calculate the volume of the tank.

Answer(b)(i) .......................................... cm3 [3]

(ii) Write your answer to part (b)(i) in litres.

Answer(b)(ii) ........................................ litres [1]

(c) The 180 000 cm3 of water flows from the tank in part (a) into the tank in part (b) at a rate of 15 cm3/s.

Calculate the time this takes. Give your answer in hours and minutes.

Answer(c) ................ h ................ min [3]

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(d)

h cm

D C

F E

A B50 cm

70 cm

x cm NOT TOSCALE

40 cm

The 180 000 cm3 of water reaches the level EF as shown above. EF = x cm and the height of the water is h cm.

(i) Using the properties of similar triangles, show that h = 2(x – 50).

Answer(d)(i)

[2]

(ii) Using h = 2(x – 50), show that the shaded area, in cm2, is x2 – 2500.

Answer(d)(ii)

[1]

(iii) Find the value of x.

Answer(d)(iii) x = ................................................. [2]

(iv) Find the value of h.

Answer(d)(iv) h = ................................................. [1]

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9 P = 2

1

3

4c m Q =

1

0

2

3c m R =

uv

0

1c m S =

w8

3

2c m

(a) Work out PQ.

Answer(a) f p [2]

(b) Find Q –1.

Answer(b) f p [2]

(c) PR = RP

Find the value of u and the value of v.

Answer(c) u = .................................................

v = ................................................. [3]

(d) The determinant of S is 0.

Find the value of w.

Answer(d) w = ................................................. [2]

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10 f(x) = 2x – 1 g(x) = x2 + x ( )h x x2

= , x 0!

(a) Find ff(3).

Answer(a) ................................................. [2]

(b) Find gf(x), giving your answer in its simplest form.

Answer(b) ................................................. [3]

(c) Find f –1(x).

Answer(c) f –1(x) = ................................................. [2]

(d) Find h(x) + h(x + 2), giving your answer as a single fraction.

Answer(d) ................................................. [4]

Question 11 is printed on the next page.

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To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after the live examination series.

Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

11 The first four terms of sequences A, B, C and D are shown in the table.

Sequence 1st term 2nd term 3rd term 4th term 5th term nth term

A3

1

4

2

5

3

6

4

B 3 4 5 6

C –1 0 1 2

D –3 0 5 12

(a) Complete the table.

[8]

(b) Which term in sequence A is equal to 37

36 ?

Answer(b) ................................................. [2]

(c) Which term in sequence D is equal to 725?

Answer(c) ................................................. [2]